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ABSTRACTS OF SHORT TALKS
REPRESENTATION OF NUMBERS BY CASCADES
c. C. Chen
Nanyang University
A cascade C is defined as a sum of binomial coeffi-
where ah > ah_ 1 > ... > at. In this expression, we assume
that ( ha \, J = 0 whenever a < h. Given a cascade C and a
sequence£ = <£h,£h_ 1 , ... ,£t> of signs (i.e. £i = +1 or -1
for each i), we define
Also, we put
aC = ( ah) +
h+1 (
ah-1) \ at) + ••• +
h t+1
We shall prove that for any sequence <no,nl,••• ,n >of inte-s gers, there exist a cascaJe C and a corresponding sequence
£of signs such that ni = £aic fori= 0,1, ... ,s where a°C =
-- 1 2 1 n n-1 C, a C = aC, a C = a(a C), and recursively, a C = a(a C).
This paper is written jointly by C. C. Chen and D. E.
Daykin and published in the Proceedings of the American
Mathematical- Society, Volume 59, Number 2, September 1976.
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NUMERICAL SOLUTION Of A TWO-POINT BOUNDARY VALUE
PROBLEM USING SPLINES
S. J. Wilson
University of Singapore
In the study of the interaction of radiation with
temperature in interstellar medium, one obtains a pair of
second order non-linear ordinary differential equations
with conditions at the two end points. This two-point boun
dary value problem is converted into a minimization problem
with linear constraints, using cubic splines and introduc
ing an error function. The Goldfarb-Lapidus gradient pro
jection algorithm.is then used to solve the minimization
problem. This method is an adaptation of Sirisena's tech
nique for solving two point boundary-value problems (J.
Optimization Theoroy and Applications, Vol. 16, 1975).
This work is done jointly with Prof. K. K. Sen and
Dr. A. N. Poo.
CONVOLUTION FUNCTIONS ON COMPACT ABELIAN GROUPS
Leonard Y. H. Yap
University of Singapore
Let G be an infinite compact abelian group with
character group G. For r > 0, define A (G) to be the set r A
of all f E Lt(G) such that the Fourier transform f E ~ (G). r
For r > 0 and sA> O, let A(r,s)(G) be the set of all
f E Lt(G) with f belonging to the Lorentz space ~(r,s)(G).
THEOREM 1. Let 1 < p ~ 2, 1 < q ~ 2 and 'let 1/ro =
1/p + 1/q -1. Then L (G)* L (G)CA ,(G), 1/ro + 1/ro' = 1, p q r
and equality holds if and only if p = q = 2.
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THEOREM 2. Let pJ qJ r be as in Theorem 1 and Zet 1/s
= 1/ p + 1/ q. Then we have
(i) there exist f £ Lp{G)J h £ Lq{G) such that f * h
~ A (SJy)(G) for aZZ S < r' and aZZ t > 0; and
( ii ) if 0 < s < sJ then there exist f £ L (G)J h £ L (G) 0 p q
such that f * h ~ A(r'Js )(G). 0
A corollary of Theorem 2 shows that Young's inequality
is the best possible in some sense. A result of R. L. Lipsman
(Duke Math. J. 36(1969)J 765-780) and a theorem of U. B.
Tewari and A. K. Gupta (BuZZ. Austral. Math. Soc. 9(1973)J
73-82) are immediate consequences of Theorem 2.
REPRESENTATIONS OF LINEAR NILPOTENT GROUPS OF CLASS TWO
H. N. Ng
University of Singapore
An irreducible complex representation of a nilpotent
group G of class two is faithful if and only if it has
degree G : 'L_Z(G)]~. A full account of the representations
of such a group will be given.
A PRESENTATION OF A GROUP
M. J. Wicks
University of Singapore
A presentation, comprising a generating set and a
set of defining relations, is obtained for r the automor
phism group of the free group of rank two. The group r
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is related to the group of non-singular transformations of
a (Euclidean) vector space. The generators of r correspond
to the elementary transformations of rotation, reflection
and shear. The defining relations allow a given automor
phism to be expressed as a product of a minimum number of
shears. The standard form is not unique but it has further
interesting properties.
ON A CONJECTURE OF ERVIN FRIED
H. P. Yap
.University of Singapore
A directed, simple graph G of order ~ 3 is said to
satisfy Fried's conditions if any two distinct vertices of
G have a unique upper bound and a unique lower bound.
Ervin Fried conjectured that if G is finite and G satisfies
Fried's conditions, then G has a triangle. E. C. Milner
proved that if G is finite and that G satisfies Fried's
conditions such that G is triangle-free, then G is regular
and hence by a result of H. H. Teh, G is a quasi-group
graph. We now prove that there is no finite directed
abelian-group graph G satisfying Fried's conditions such
that G is triangle-free. We further prove that if G is
a finite directed graph satisfying Fried's conditions
and that G is triangle-free, then the order of G is ~ 31.
THE HYPERCENTRE OF A FINITE GROUP
Peng Tsu Ann
University of Singapore
Let G be a group. Let
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be the upper central series of G (i.e., Z1(G) is the centre
of G and z.+1
(G)/Z.(G) = Zj(G/Z.(G)) fori;::. 0). The l l l
terminal member of the above series is called the hypercentre
of G. We report the following result on the hypercentre of
a finite group.
TEEOREM. Let G be a finite group. A au /:;group X of
G l i es i n th e h y p er c en t r e of G if and o n ly if X n S l i es
in the hypercentre of S for eVery soluble aul;group 5 of G.
DESIGN OF DISTURBANCES ABSORBING CONTROLLERS FOR
LINEAR STOCHASTIC SYSTEMS
N. K. Loh
University of Iowa and University of Malaya
In this paper, techniques have been developed for
rejecting, utilizing, or minimizing the effects of internal
and external disturbances in linear control systems. The
disturbances are assumed to have known waveform structure,
such as random step functions, random ramp functions,
random sinusoidal functions, or more complicated functions;
but their instantaneous amplitudes, phases, etc. are random
and unknown. No statistical information about the disturbances
is required and/or used. The resulting control system is
disturbance-adaptive in nature, so that it has marked impro
vement in its performance when compared with a control system
designed under the conventional methods. The design tech
niques developed have been successfully applied to the
design of various real world control systems, some of which
are shown in this paper.
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AN INTE·3RAL EQl:A':'ION WITH A PROBABILISTIC APPLICATION
Louis H. Y. Chen
University of Singapore
Let v be a real-valued, bounded and Borel measurable
function defined on [O,oo), ~be a real measure defined
on the Borel subsets of [o,oo) but with no atom at zero,
and A be a positive number. Define S = eA(~- 6 ) where A,~
6 is the Dirac measure at zero. It is proved that there
exists a real-valued, Borel measurable function f defined
on [o ,"'), with sup I wf (w) I < oo and satisfying the integral w~O
equation
A Jtf(w+t)d~(t) = v(w), wf(w)
if and only if JvdS>,v " 0. It is also proved that the
solution f is unique except at zero. Furthermore, an
explicit expression for f and an upper bound for suplwf(w)l w~O
are obtained. The integral equation is then used to obtain
a necessary and sufficient condition for the distributions of
the row sums of an infinitesimal, finitely dependent trian
gular array of non-negative random variables to converge
in total variation to the real measure S, . 1\.~
THE MINIMAL a-DEGREE PROBLEM
C. T. Chong
University of Singapore
For each admissible ordinal a, a set T C. a is of
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-
minimal a-degree if every set of strictly lower a-degree
is a-recursive. One of the most important areas of study
in higher recursion theory (cf. Sacks [3] ) is that of
minimal degrees in models of weak and strong set theories.
The study is important as on the one hand it leads to the
theory of imbeddings of lattices into degrees - which in
turn yields undecidability results for various theories -
and on the other hand it leads to the discovery by Sacks
of the technique of forcing with perfect closed sets, a
useful tool in studying independence results in set theory.
The first theorem on minimal a-degrees was proved by Spector
[6] on a=w, refined by Sacks [4], and lifted to countable
admissible ordinals by Macintyre (2]. Shore [s] proved
the existence of minimal a-degrees for all E2-admissible
ordinals a, and recently Maass [1] improved this result
to the case o2p(a) ~ o2cf(a). For various reasons, the
remaining case stays insurmountable. In this talk we
state a characterization theorem (via the notion of
genericity) which we believe implies that minimal a-degrees
do not exist for some a's.
THEOREM. Assume that a is not E2 -admissible. A
regular, hyperregular set T is of minimal a-degree if
and only if it is generic.
References
1 W. Maass, "On minimal pairs and minimal degrees in
higher recursion theory," preprint, M .. I.T., 1976.
2 J. Macintyre, "Minimal a-recursion theoretic degrees",
J. Symrolic Logic, 38(1973), 18-28.
3 G. Sacks, Higher recul'S ion th eo:ry, Springer Verlag,
to appear.
4 G. Sacks, "On the degrees less than 0 1 ", Annals of
Math . , 7 7 (19 6 3 ) , 211-2 31.
5 R. Shore, "Minimal a-degrees", Annals of Math. Logic,
4(1972), 393-414.
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6 C. Spector, "On degrGes of recursive unsolvability",
Annals of Matl-t., 64(1956), 581-592.
THE FUNDAMENTAL GROUPOID OF A SPACE ASSOCIATED WITH A COVER
Abdul Razak bin Salleh
Universiti Kebangsaan Malaysia
Let U be any cover of a topological space X. We
construct a classifying space BU associated with the cover
U. This space has been considered by G. Segal who has
proved that the projection p : BU ~ X is a homotopy equi
valence if U is Bumerable. This of course implies that
np nBU ~ nX is an equivalence of fundamental groupoids
if U is numerable. We have proved that p induces an
equivalence of fundamental groupoids when the interiors
of the elements of U cover X and have showed how this is
related to a theorem of Macbeath-Swan relating n 1 (G) to
G in the case when group G acts on X.
This is part of the author's Ph.D thesis submitted
to the University of Wales in December 1976.
THE LINEAR AUTOMORPHISMS OF CERTAIN IRREDUCIBLE LINEAR GROUPS
Y. K. Leong
University of Singapore
Let G be a subgroup of the general linear group GL(r,F)
over some field F which is splitting for G. The group of
linear automorphisms of G is defined by
lin aut G = {~ £ aut G : there exists y £ GL(r,F) -I
such that x~ = y xy for all x £ G},
and the group of inner automorphisms of G is defined by
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inn G = {~ £ aut G : there exists y £ G such that
x~ = y- 1 xy for all x £ G}.
We study the structure of lin aut G/ inn G for certain
finite q-groups of class 2 with cyclic centre, where q is
an odd prime, and extend some results of David L. Winter
on the automorphism group of an extraspecial p-group.
A CHARACTERISATION OF FINITE SIMPLE GROUPS
WITH A COMPONENT ISOMORPHIC TO PSL3(4)
Cheng Kai Nah
University of Singapore '
It is well-known that the finite simple groups con
sti t ute the fundamental building blocks of all the finite
groups. In order to understand groups better, one would
naturally like to have a complete list of all finite
simple groups. Hitherto, almost every finite simple group
appears in one of the following 3 infinite families:
(l)
( 2 )
( 3)
cyclic groups of prime order,
alternating groups A of degree n with n ~ 5, n groups of Lie type.
Besides these, there are so far 26 exceptions- they are the
so-called sporadic simple groups. Whether there are finitely
many sporadic simple groups or whether they belong in one
way or another to some infinite families remains an open
question. Through the characterisatio n of simple groups
one attempts to settle this problem.
The general characterisation problem is as follows:
Let (X) be a given group property. Determine all finite
non-abelian simple groups which satisfy property (X).
Recently a systematic approach towards this problem
has been developed and we are led to a study of simple
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groups in two categories: (1) groups of component-type,
(2) groups of non component-type.
A characterisation of groups of component-type by the exis
tence of certain "standard subgroups" was introduced by
Aschbacher.
In this connexion, I have proved the following:
THEOREM. Let G be a finite simple non-abelian group
which possesses a standard subgroup A such that AjZ(A) is
isomorphic to PSL3(4). Assume that 2 10 TjG\. Then G is
isomorphic to He or O'N.
MULTIPLIERS FROM L1 TO A BANACH L1-MODULE
Quek Tong Seng and Leonard Y. H. Yap
University of Singapore
Let G be a locally compact abelian group with Haar
measure A. Let L , 1 ~ p ~ ~ , denote the usual Lebesgue p
space with respect to A. Let A C L , p
Banach L1-module such that II lip < II
l ~ p < ~ , be a
IIA' and let {ect}
be a net in L1 such that II ea [\ 1 = 1 for all a and {ea} is
a common approximate identity for L1, L p and A. Define
the relative completion A of A to be the space
A = {f E L sup II f * ect II A < ~ } p
ct
with the norm II f \I;. = sup II f "'
e IIA' ct
THEOREM. (L1,A), the s paoe of multipliers from £1 -to A, is is orne trio ally isomorph io to A if p > 1, or if p = 1 and (LI,A) c L 1.
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This theorem and some of its consequences will be
d iscussed.
A CONVOLUTION ALGEBRA OF OPERATORS
Stephen T. L. Choy
University of Singapore
Let S be a compact semigroup and let A be a unital
Banach algebra. Denote C(S,A) the space of continuous
functions from S to A with uniform norm. Let T be a
bounded linear operator from C(S,A) to A. It is shown
in this paper that the set W(S,A) of all weakly compact
operators T which are multipliers of A can be formed as a
Banach algebra with convolution operation. Furthermore if
KCS,A) is the set of compact operators T which are multi
pliers of A, then the maximal ideal space 6K(S,A) is
6(C 1 (S)) x 6(M(A)), the Cartesian product of the maximal
ideal spaces of the measure algebra C 1 (S) and the algebra
M(A) of the multipliers of A.
A TABULAR ALGORITHM FOR FINDING THE PARTIAL QUOTIENTS OF
CONTINUED FRACTIONS EQUIVALENT TO y = IN
A. D. Villanueva
Pappacon Engineers & Associates
Research on continued fractions has revealed many
interesting facets of the irrational number y = IN where
N is a rational positive intege~. For example:-
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(1) y = IN = 1M 2 +2M-l = [M,l,(M-1)1,2MJ ' where M > 1.
Thus, y = IN = h4 = l5 2+2x5-l = [5,1,4,1,1oJ for M = 5.
( 2) y = IN = I( 211+ 1) 2 + (3M+ 2) = (C2M+l),l,2,1,2(2M+l)]
where M ~ 1. Thus, y = 195 = IC2x4+1) 2 +(3x4+1) = [9 '1 '2 '1 '18 J where M = 4.
[c 6M + 3 ) , H , 1 , 1 , ( 3 H + 1 ) ( 4 M + 2 )
~ 1. Thus, y = lffi = (3) y = IN= 1(6!'-1+3) 2 +12 =
(3M+l),1,l,M,2(6M+3)J where M
1(6x2+3) 2 +12 = [15,2,1,1,7,10,7,1,1,2,30] for H = 2.
There are literally thousands of these unique
relations that can only be found by constructing a table of
partial fractions.for y = IN and comparing the behaviour
of particular groups of partial quotients that appear to
form a pattern - by the analytic method. A synthetic
approach is difficult if not impossible.
The tabular method presented is a simplified arith
metical algorithm that dispenses with the arduous and
time-consuming algebraic algorithm that has to be written
down and evaluated, term by term, in the conventional
method of finding the partial quotients of y = IN.
* * * * * * * * * * * * *
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'